Overview. Efficient Simplification of Point-sampled Surfaces. Introduction. Introduction. Neighborhood. Local Surface Analysis

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1 Overview Efficient Simplification of Pointsampled Surfaces Introduction Local surface analysis Simplification methods Error measurement Comparison PointBased Computer Graphics Mark Pauly PointBased Computer Graphics Mark Pauly Introduction Pointbased models are often sampled very densely Many applications require coarser approximations, e.g. for efficient Storage Transmission Processing Rendering we need simplification methods for reducing the complexity of pointbased surfaces Introduction We transfer different simplification methods from triangle meshes to point clouds: Incremental clustering Hierarchical clustering Iterative simplification Particle simulation Depending on the intended use, each method has its pros and cons (see comparison) PointBased Computer Graphics Mark Pauly 3 PointBased Computer Graphics Mark Pauly 4 Local Surface Analysis Cloud of point samples describes underlying (manifold) surface We need: mechanisms for locally approximating the surface MLS approach fast estimation of tangent plane and curvature principal component analysis of local neighborhood Neighborhood No explicit connectivity between samples (as with triangle meshes) Replace geodesic proximity with spatial proximity (requires sufficiently high sampling density!) Compute neighborhood according to Euclidean distance PointBased Computer Graphics Mark Pauly 5 PointBased Computer Graphics Mark Pauly 6

2 Neighborhood knearest neighbors Neighborhood Improvement: angle criterion (Linsen) can be quickly computed using spatial datastructures (e.g. kdtree, octree, bsptree) requires isotropic point distribution project points onto tangent plane sort neighbors according to angle include more points if angle between subsequent points is above some threshold PointBased Computer Graphics Mark Pauly 7 PointBased Computer Graphics Mark Pauly 8 Neighborhood Local Delaunay triangulation (Floater) Covariance Analysis Covariance matrix of local neighborhood N: pi p pi p C = L L, pi p n pi p n T i j N project points into tangent plane compute local Voronoi diagram with centroid p = N p i i N PointBased Computer Graphics Mark Pauly 9 PointBased Computer Graphics Mark Pauly 0 Covariance Analysis Consider the eigenproblem: C v l = λ v, l {0,,} l C is a 3x3, positive semidefinite matrix All eigenvalues are realvalued The eigenvector with smallest eigenvalue defines the leastsquares plane through the points in the neighborhood, i.e. approximates the surface normal l Covariance Analysis The total variation is given as: i N We define surface variation as: λ0 σ n ( p) =, λ0 λ λ λ λ λ 0 p i p = λ λ λ measures the fraction of variation along the surface normal, i.e. quantifies how strong the surface deviates from the tangent plane estimate for curvature 0 PointBased Computer Graphics Mark Pauly PointBased Computer Graphics Mark Pauly

3 Covariance Analysis Comparison with curvature: Surface Simplification Incremental clustering Hierarchical clustering Iterative simplification Particle simulation original mean curvature variation n=0 variation n=50 PointBased Computer Graphics Mark Pauly 3 PointBased Computer Graphics Mark Pauly 4 Incremental Clustering Clustering by regiongrowing: Start with random seed point Successively add nearest points to cluster until cluster reaches maximum size Choose new seed from remaining points Growth of clusters can also be bounded by surface variation Curvature adaptive clustering Incremental Clustering Incremental growth leads to internal fragmentation assign stray samples to closest cluster Note: this can increase maximum size and variation bounds! PointBased Computer Graphics Mark Pauly 5 PointBased Computer Graphics Mark Pauly 6 Incremental Clustering Replace each cluster by its centroid Hierarchical Clustering Topdown approach using binary space partition: Split the point cloud if: Size is larger than userspecified maximum or Surface variation is above maximum threshold original model with colorcoded clusters (34,384 points) (,000 points) Split plane defined by centroid and axis of greatest variation (= eigenvector of covariance matrix with largest associated eigenvector) Leaf nodes of the tree correspond to clusters PointBased Computer Graphics Mark Pauly 7 PointBased Computer Graphics Mark Pauly 8 3

4 Hierarchical Clustering D example Hierarchical Clustering Adaptive clustering original model with colorcoded clusters (34,384 points) (,000 points) PointBased Computer Graphics Mark Pauly 9 PointBased Computer Graphics Mark Pauly 0 Iterative Simplification Iteratively contracts point pairs Each contraction reduces the number of points by one Contractions are arranged in priority queue according to quadric error metric (Garland and Heckbert) Quadric measures cost of contraction and determines optimal position for contracted sample Equivalent to QSlim except for definition of approximating planes Iterative Simplification Quadric measures the squared distance to a set of planes defined over edges of neighborhood plane spanned by vectors e p p and e e e e = i = n n p p i PointBased Computer Graphics Mark Pauly PointBased Computer Graphics Mark Pauly Iterative Simplification Particle Simulation Resample surface by distributing particles on the surface Particles move on surface according to interparticle repelling forces Particle relaxation terminates when equilibrium is reached (requires damping) Can also be used for upsampling! original model (87,664 points) (,000 points) remaining point pair contraction candidates PointBased Computer Graphics Mark Pauly 3 PointBased Computer Graphics Mark Pauly 4 4

5 Particle Simulation Initialization randomly spread particles Repulsion linear repulsion force Fi ( p) = k( r p pi ) ( p pi ) only need to consider neighborhood of radius r Projection keep particles on surface by projecting onto tangent plane of closest point apply full MLS projection at end of simulation PointBased Computer Graphics Mark Pauly 5 Particle Simulation Adaptive simulation Adjust repulsion radius according to surface variation more samples in regions of high variation original model (75,78 points) (6,000 points) PointBased Computer Graphics Mark Pauly 6 Particle Simulation Usercontrolled simulation Adjust repulsion radius according to user input Measuring Error Measure the distance between two pointsampled surfaces using a sampling approach Maximum error: max ( S, S ) = max d(, S q Q q ) Twosided Hausdorff distance Mean error: S S = d q S avg (, ) (, ) Q q Q Areaweighted integral of pointtosurface distances Q is an upsampled version of the point cloud that describes the surface S PointBased Computer Graphics Mark Pauly 7 PointBased Computer Graphics Mark Pauly 8 Measuring Error d ( q, S ) measures the distance of point q to surface S using the MLS projection operator with linear basis functions Comparison Error estimate for Michelangelo s David simplified from,000,000 points to 5,000 points PointBased Computer Graphics Mark Pauly 9 PointBased Computer Graphics Mark Pauly 30 5

6 Comparison Execution time as a function of target model size (input: dragon, 535,545 points) Comparison Execution time as a function of input model size (reduction to %) PointBased Computer Graphics Mark Pauly 3 PointBased Computer Graphics Mark Pauly 3 Comparison Summary Pointbased vs. Mesh Simplification Incremental Clustering Hierarchical Clustering Iterative Simplification Particle Simulation Efficiency o Surface Error Control o Implementation o pointbased simplification with subsequent mesh reconstruction mesh reconstruction with subsequent mesh simplification (QSlim) pointbased simplification saves an expensive surface reconstruction on the dense point cloud! PointBased Computer Graphics Mark Pauly 33 PointBased Computer Graphics Mark Pauly 34 References Pauly, Gross: Efficient Simplification of Pointsampled Surfaces, IEEE Visualization 00 Shaffer, Garland: Efficient Adaptive Simplification of Massive Meshes, IEEE Visualization 00 Garland, Heckbert: Surface Simplification using Quadric Error Metrics, SIGGRAPH 997 Turk: ReTiling Polygonal Surfaces, SIGGRAPH 99 Alexa et al. Point Set Surfaces, IEEE Visualization 00 PointBased Computer Graphics Mark Pauly 35 6